Let \(X\) be a scheme over a ring \(A\), \(\mcl\) an invertible sheaf on \(X\), and \(s_0, \dots, s_n \in H^0(X, \mcl)\) sections that generate \(\mcl\). Then there is a unique \(A\)-morphism \[ \varphi : X \to \PP^n_A, \qquad p \mapsto [s_0(p) : \cdots : s_n(p)] , \] with \(\mcl \cong \varphi^* \OO_{\PP^n}(1)\) and \(s_i = \varphi^*(x_i)\) under this isomorphism. Conversely, every \(A\)-morphism \(\varphi : X \to \PP^n_A\) arises this way, from \(\mcl = \varphi^*\OO(1)\) and \(s_i = \varphi^*(x_i)\). [@Har10a, Theorem II.7.1]
(Linear systems with base points)
Let \(X\) be a \(k\)-scheme, \(V \subseteq H^0(X, \mcl)\) a subspace with basis \(s_0, \dots, s_n\), and \(B = \bigcap_i V(s_i)\) its base locus. The sections restricted to the open set \(X \sm B\) generate \(\mcl\) there, so they define a morphism \(\varphi_V : X \sm B \to \PP^n_k\), which is \(\PP V\) up to the choice of basis. On the open set \(D(s_i) \subseteq X \sm B\) where \(s_i\) does not vanish, \(\varphi_V\) maps into the standard chart \(D(x_i)\) and is given by \(x_j/x_i \mapsto s_j/s_i\).
(Degrees on curves)
Let \(X\) be a nonsingular projective curve over \(k = \bar k\) and \(V \subseteq H^0(X, \mcl)\) base-point free.
If \(\dim V = 2\), the morphism \(\varphi_V : X \to \PP^1\) is finite of degree \(\deg \mcl\).
If \(\mcl\) is very ample and \(V = H^0(X, \mcl)\), then \(\varphi_V\) embeds \(X\) as a curve of degree \(\deg \mcl\) in \(\PP^n\), and \(\mcl \cong \varphi_V^* \OO(1)\).
[@Har10a, Proposition II.6.9, Theorem II.7.1]
Erratum: the source states the degree of the pencil map without assuming \(V\) base-point free. If a two-dimensional \(V\) has base points, the fixed part \(F\) of the pencil is an effective divisor of positive degree, and the morphism \(X \to \PP^1\) that \(V\) induces has degree \(\deg \mcl - \deg F\).
(Closed immersion criterion)
Let \(X\) be a projective scheme over \(k = \bar k\) and \(V \subseteq H^0(X, \mcl)\) base-point free. \(\varphi_V\) is a closed immersion exactly when the system separates points and separates tangent vectors:
for all closed points \(p \neq q\) there is \(D \in \abs{V}\) with \(p \in \supp D\) and \(q \notin \supp D\);
for every closed point \(p\) and every \(0 \neq t \in T_p X\) there is \(D \in \abs{V}\) with \(p \in \supp D\) and \(t \notin T_p(D)\), where \(D\) is regarded as a closed subscheme.
[@Har10a, Proposition II.7.3]
Base-point freeness is exactly what makes the formula define a point of \(\PP^n\) at all: some coordinate must be nonzero. The two separation conditions are then injectivity and injectivity on tangent spaces, which is what a closed immersion is.
The choice of \(V\) matters, not just \(\mcl\): taking the complete system gives the largest target, and a proper subspace gives a linear projection of that image. Standard examples are \(\abs{\OO_{\PP^1}(d)}\), which embeds \(\PP^1\) as the rational normal curve in \(\PP^d\), and \(\abs{\OO_{\PP^n}(2)}\), which gives the Veronese embedding. On a curve the two separation conditions become the numerical statement about \(\ell(D - p - q)\).
Dependencies
D-DIVLINSYSLinear systems, base loci, and base-point freenessuses
Backlinks
D-DIVAMPLEAmple and very ample divisorsuses